Question

(a) Place the following gases in order of increasing average molecular speed at \(25^{\circ} \mathrm{C}: \mathrm{Ne}, \mathrm{HBr}, \mathrm{SO}_{2}, \mathrm{NF}_{3}, \mathrm{CO}\). (b) Calculate the rms speed of \(\mathrm{NF}_{3}\) molecules at \(25^{\circ} \mathrm{C}\).

Short answer

The order of gases based on increasing average molecular speed at 25°C is HBr < NF₃ < SO₂ < CO < Ne. The rms speed of NF₃ molecules at 25°C is 382.13 m/s.

Step-by-step solution

(a) Formula for Average Molecular Speed

For any gas, the formula for the average molecular speed (u) is given by: \[u = \sqrt{\frac{8RT}{\pi M}}\] where R is the gas constant = 8.314 J/(mol·K), T is the temperature in Kelvin, and M is the molar mass of the gas in kg/mol. Given that the temperature is 25°C, which is 298.15 K.

Molar Masses of Gases

We need the molar masses of each gas in kg/mol. Ne = 20.18 g/mol = 0.02018 kg/mol HBr = 80.91 g/mol = 0.08091 kg/mol SO₂ = 64.07 g/mol = 0.06407 kg/mol NF₃ = 71.0 g/mol = 0.0710 kg/mol CO = 28.01 g/mol = 0.02801 kg/mol Now, we can find the average molecular speeds of each gas.

Average Molecular Speeds

Using the formula for average molecular speed, we calculate the speeds for each given gas: u(Ne) = \(\sqrt{\frac{8(8.314)(298.15)}{\pi (0.02018)}}\) ≈ 605.41 m/s u(HBr) = \(\sqrt{\frac{8(8.314)(298.15)}{\pi (0.08091)}}\) ≈ 322.82 m/s u(SO₂) = \(\sqrt{\frac{8(8.314)(298.15)}{\pi (0.06407)}}\) ≈ 358.84 m/s u(NF₃) = \(\sqrt{\frac{8(8.314)(298.15)}{\pi (0.0710)}}\) ≈ 349.44 m/s u(CO) = \(\sqrt{\frac{8(8.314)(298.15)}{\pi (0.02801)}}\) ≈ 440.13 m/s

Order of Gases

Now we can put the gases in order of increasing average molecular speed: HBr < NF₃ < SO₂ < CO < Ne (answer for part a)

(b) Formula for RMS Speed

For any gas, the formula for root-mean-square speed (v_rms) is given by: \[v_{rms} = \sqrt{\frac{3RT}{M}}\] We'll use this formula to find the rms speed of NF₃ at 25°C.

RMS Speed of NF₃

Given that the temperature is 25°C (298.15 K) and the molar mass of NF₃ is 0.0710 kg/mol, the rms speed is: v_rms(NF₃) = \(\sqrt{\frac{3(8.314)(298.15)}{(0.0710)}}\) ≈ 382.13 m/s (answer for part b)

Key concepts

Average Molecular Speed

In the realm of chemistry and physics, the average molecular speed is a vital concept that reflects the typical velocity of molecules within a gaseous sample. Since the true behavior of gas molecules includes a vast array of individual speeds, the average provides a simplified yet useful representation of their kinetic activity.

To understand this concept, imagine billions of gas particles zipping around in all directions at diverse speeds. Some collide, others dodge, and with every passing moment, their speeds change. The average molecular speed, denoted as 'u', is the mean speed of these particles and can be calculated using the fundamental formula:
\[\[\begin{align*}u = \textstyle\sqrt{\frac{8RT}{\pi M}}\end{align*}\]\]where 'R' stands for the universal gas constant, 'T' is the absolute temperature in Kelvin, and 'M' is the molar mass.

• R (The gas constant): A cornerstone in thermodynamic equations, R accounts for the physical properties of gases in the equation.
• T (Absolute temperature): It sets the scene for molecular activity. Higher temperatures lead to greater average speeds.
• M (Molar mass): The average weight of one mole of gas molecules. A pivotal factor, as lighter gases have higher average speeds at the same temperature.

During problem-solving, students often make mistakes with unit conversions or temperature adjustments, overlooking the need to convert Celsius to Kelvin or grams to kilograms when evaluating molar mass. Careful attention to these details ensures accurate calculation of average molecular speeds.

RMS Speed

Switching gears from the average, let's delve into the rms speed—root-mean-square speed, a precise gauge of gas molecules' kinetic activity. The rms speed, symbolized as 'v_rms', captures the square root of the mean of the squares of the speeds of a collection of particles.

This pivotal thermodynamic measure tells us about the energy within a gas sample, through the formula:
\[\[\begin{align*}v_{\text{rms}} = \sqrt{\frac{3RT}{M}}\end{align*}\]\]The elegance of 'v_rms' lies in its ability to factor in the distribution of velocities, offering a concrete reflection of the intrinsic energy in the gas. To avoid confusion, always remember:

• The RMS speed isn't identical to the average speed - while they both describe molecular motion, rms considers the distribution of all molecular speeds.
• The RMS value will generally be higher than the average speed for a given temperature and molar mass, due to the squaring of speeds in its calculation.

The rms speed jumps into action, for instance, when calculating the diffusion rate of gases or theoretical modeling of gas behavior at varying temperatures and molar masses. An intricate understanding of this principle is not only academically rewarding but practically beneficial in fields such as meteorology, aeronautics, and material science.

Molar Mass

The molar mass is a cornerstone concept in the study of chemistry and provides a bridge between the microscopic world of atoms and molecules and the macroscopic world we can measure. Represented by 'M' and typically expressed in grams per mole (g/mol), molar mass answers the question of how much one mole of a substance weighs.

Molar mass directly influences other properties like average molecular speed and rms speed, acting as a denominator that inversely affects these values - the higher the molar mass, the slower the molecular speeds.

Molar Mass in Calculations

While tackling exercises involving molecular speed, the molar mass ties directly into the kinetic molecular theory equation. Remembering to convert the molar mass from grams per mole to kilograms per mole (kg/mol) when using SI units is a frequent stumbling block for students.

• Conversions are crucial: 1 g/mol = 0.001 kg/mol. No room for error here if correct results are to be obtained.
• The impact on gas behavior: A heavier molar mass means gases move more sluggishly at the same temperature versus their lighter counterparts.

In practical applications, understanding molar mass can assist in determining the composition of unknown substances or the reaction yields in chemical processes. It’s an indispensable tool for scientists and students alike, not only in theoretical exercises but also in real-world measurements and industrial applications.